Space Situational Awareness (SSA) includes knowledge of the near-Earth space environment which can be accomplished through the tracking and identification of Earth-orbiting space objects to protect space assets and maintain awareness of potentially adversarial space deployments. Although current operational systems have performed well, future needs will far exceed current capabilities. With the instantiation of more accurate sensors and the increased probability of future collisions between space objects, the potential number of newly discovered objects is likely to increase by an order of magnitude within the next decade, thereby placing an ever increasing burden on current operational systems. Moving forward, the implementation of new, innovative, rigorous, robust, and autonomous methods for space object identification and discrimination are required to enable the development and maintenance of the present and future space catalog and to support the overall SSA mission.
Fundamental to the success of the SSA mission is the rigorous inclusion of uncertainty in the space surveillance network (SSN). The proper characterization of uncertainty, sometimes called covariance realism, is a common requirement to many SSA functions including tracking and data association, resolution of uncorrelated tracks (UCTs), conjunction analysis and probability of collision, sensor resource management, and anomaly detection. While tracking environments, such as air and missile defense, make extensive use of Gaussian and local linearity assumptions within algorithms for uncertainty management, space surveillance is inherently different due to long time gaps between updates, high mis-detection rates, non-linear and non-conservative dynamics, and non-Gaussian phenomena. What is state-of-the-art and robust for air and missile defense need not be applicable to SSA.
The field of sequential state estimation has much of its origins in the pioneering work of R. E. Kalman. Considered to be one of the most simple dynamic Bayesian networks, the Kalman filter updates a system state recursively over time using incoming measurements and mathematical process models. The basic Kalman filter assumes linearity in the underlying dynamical and measurement models and that all error terms are Gaussian. At the other end of the spectrum is the general Bayesian non-linear filter which updates the full Probability Density Function (PDF) of the system recursively over time while permitting non-Gaussian error terms and non-linear process models. Between the Kalman filter and the general Bayesian framework are a host of sub-optimal methods for sequential filtering which have been developed over the past fifty years and tailored to specific applications. The most common extensions and generalizations of the Kalman filter include the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) which both work on non-linear systems. A feature of the latter is its “derivative-free” nature. Within the filter prediction step, the propagated state estimate and covariance are reconstructed from a deterministically chosen set of “sigma points” propagated through the full non-linear dynamics. The equivalence between this reconstruction or “unscented transform” with Gauss-Hermite quadrature has been established. Variations and extensions of the UKF, including a more numerically stable “square-root” version and the higher-order Gauss-Hermite filters, have been formulated. In space surveillance, the state or orbital uncertainty can be highly non-Gaussian and filtering techniques beyond the EKF and UKF are sometimes required. Examples include Gaussian sum (mixture) filters, filters based on non-linear propagation of uncertainty using Taylor series expansions of the solution flow, particle filters, and the probability hypothesis density filter and its generalization, the cardinalized probability hypothesis density filter.
A drawback of many existing methods for non-linear filtering and uncertainty management, including the ones reviewed above, is the constraint that the state space be defined on an n-dimensional Cartesian space n. Any statistically rigorous treatment of uncertainty uses PDFs defined on the underlying manifold on which the system state is defined. In the space surveillance tracking problem, the system state is often defined with respect to orbital element coordinates. In these coordinates, five of the six elements are approximated as unbounded Cartesian coordinates on 5 while the sixth element is an angular coordinate defined on the circle  with the angles θ and θ+2πk (where k is any integer) identified as equivalent (i.e., they describe the same location on the orbit). Thus, more rigorously, an orbital element state space is defined on the six-dimensional cylinder 5×. Indeed, the mistreatment of an angular coordinate as an unbounded Cartesian coordinate can lead to many unexpected software faults and other dire consequences.